Impact of time-related factors on biologically accurate radiotherapy treatment planning

The cell survival fraction S is given by the linear–quadratic (LQ) model as

where α and β are the LQ model parameters and d _k is the dose of fraction k. Therefore, the Poisson LQ NTCP model based on the relative seriality model [10] and the critical volume model [11] is expressed as follows:

where M denotes the total number of voxels, d _k,i is the dose of fraction k to voxel i, and n is the total number of fractions. N ₀ denotes the initial number of cells, v _i is the relative volume of voxel i, and s represents the seriality.

To take radiation-induced cell damage into account, we use the IR model [2], which introduces the concept of “dose equivalent of IR.” The idea is that after a dose of size d, the injury induced by some fraction θ of the dose is still unrepaired by the time the next dose is given. The fraction θ is assumed to decay exponentially with time according to

where Δt is the inter-fractional interval and μ is the repair constant. Using the repair half-time T_1/2 as the time needed for half the damaged calls to be repaired, μ is expressed as

By incorporating the cell repair effect on successive doses [3], the survival fraction derived from the LQ model is given bywhere θ _q  =  exp (−μ∆t _q ), and Δt _q is the time between fractions q and q + 1. Thereafter, the NTCP including cell repair is expressed as

NTCP parameters for the spinal cord were reported as D ₅₀  = 68.6 Gy, γ = 1.90, α/β = 3.00, and a seriality of 4.00 [12], which were based on radiation tolerance data for clinical myelitis necrosis [13]. However, this NTCP model should have repair term T_1/2 potentially because this was derived from clinical fractionated irradiation. On the other study, T_1/2 was reported to be 5 h with 95% CI of 0.6 – 9.3 h, which is obtained by irradiation to rat spinal cord with variable intervals [4]. In this study, therefore, we examined the dependence of NTCP on T_1/2 varying from 0 h to 10 h to cover 95% CI on the assumption that NTCP reported in Ref. [12] was given by potentially including T_1/2 = 5 h, as following procedure.

This procedure is also described in Fig. 1.

RayStation (ver. 7.2.5, RaySearch Laboratories, Stockholm, Sweden) was used to create retrospective 3D conformal radiation therapy plans for six adult patients receiving CSI from a Siemens Artiste linear accelerator with a 6-MV photon beam, in which the dose distribution is calculated by the collapsed cone convolution superposition method [14].

For each patient, we prescribed a total dose of 36 Gy (20 fractions of 1.8 Gy each) to the target with the margin of 5 mm to both the spinal cord and the brain, in which the minimum dose was constrained not to be less than 95% of the prescribed dose. The treatment was assumed to be performed in five fractions every week without weekends, as is usual in clinical practice.

We used two opposing lateral fields to treat the entire brain and two abutted spinal fields to treat the upper and lower spinal fields [15]. The brain-field isocenter was located at the volumetric center of the brain, and we used only a longitudinal shift for the spinal-field isocenters. We used the couch angle to match the brain field with the inferior edges of the two lateral beams, and we rotated the collimator to match the upper spinal-field divergence. For the lower spinal field, we rotated the gantry to match the inferior edge divergence of the upper spinal field with a couch rotation of 90°. To facilitate several types of junction shift, we defined three different junction positions in the case of no gap (i.e., a gap size of zero) (Fig. 2). Junction position B was established to match each field exactly. Junction position A was achieved by closing two multi-leaf collimator (MLC) leaves (1 cm) of each anterior and posterior edge of the upper spinal field and by opening two MLC leaves of each posterior edge of the brain field and the anterior edge of the lower spinal field. Junction position C was achieved by moving the MLC leaves in the opposite directions of those for junction position B.

In addition to the case of no gap, gaps of 5 and − 5 mm (i.e., a 5-mm overlap) were created in the same manner. Although these are worst case scenario of systematic errors, it could arise because of setup errors or patient movement. For example, a mistake in counting a patient’s vertebrae could lead to the treatment area being located erroneously, thus resulting in overlapping junction areas [16].

Using the junction positions defined above, some junction-shift schedules were determined for each gap size assuming five fractions per week (Table 1); this process was based on Ref. [9]. Assuming two junction positions (positions A and B), biweekly, weekly, and daily plans refer to changing the junction every two weeks, every week, and every day, respectively. Assuming three junction positions (positions A, B, and C), a 1/3 treatment plan refers to changing the junction every 1/3 of the treatment (i.e., after 7 and 14 fractions). The simplest schedules among each number of junction positions (i.e., the biweekly plan for two junctions and the 1/3 treatment plan for three junctions) were taken as the reference schedules.

First, to estimate the effect of the value of T_1/2, the NTCP was calculated with the IR model via the biological evaluation tool RayBiology in RayStation, in which the junction-shift schedule was fixed to the reference schedules for each number of the junction positions.

Second, the NTCP in the case of changing the junction-shift schedule was examined to investigate whether it was affected. For each junction position, the variation in NTCP when changing from the reference schedules to the other schedules was calculated.

To examine the NTCP dependence on both T_1/2 and the dose, we calculated the NTCP manually for the spinal cord by assuming many voxels. We began by providing a uniform dose to the entire spinal cord. Thereafter, we gave a uniform dose to 1% of the volume of the spinal cord and no dose to the remaining 99%. To eliminate variations in dose per fraction under a fixed total dose, the total dose was expressed by the equivalent dose in 2 Gy fractions (EQD₂) and was varied from 30 to 120 Gy. The fractionation schedule (20 fractions/4 weeks) was the same as those for the CSI planning. Given that we assumed each voxel to be exposed to a fixed dose, no variation in junction-shift schedule was considered in this calculation.

We evaluated the statistical significance of the proposed changes in NTCP. Given the small sample size (n = 6), we used the Shapiro–Wilk test to validate the NTCP normality. Thereafter, we used a paired Student t-test to investigate the statistical significance with a confidence level of 95%.

Figure 3 shows the NTCP as a function of T_1/2 on the reference junction-shift schedules (i.e., the biweekly plan for two junction positions and the 1/3 treatment plan for three junction positions). The NTCP was largest with a gap size of − 5 mm, followed by 0 and 5 mm, although the difference between the latter two was small. With the same gap size, three junction positions resulted in a lower NTCP than two junction positions, particularly for a gap size of − 5 mm.

For all gap sizes, the NTCP with the IR model varied depending on T_1/2. In comparison to the NTCP with T_1/2 = 5 h, that with the shorter T_1/2 was significantly smaller, and that with the longer T_1/2 was significantly larger. In addition, the longer T_1/2 gave more prominent increase of NTCP, which reached a maximum with T_1/2 = 10 h.

Figure 4 shows the NTCP deviation when the junction-shift schedule was changed from the reference schedules to the other schedules for each number of junctions. For two junction positions, no NTCP reduction was observed upon changing to the weekly plan. However, for a gap size of − 5 mm, a statistically significant deviation of NTCP was seen for T_1/2 longer than 6 h upon changing to the daily plan; the NTCP deviation increased negatively as T_1/2 decreased and reached a maximum of − 0.07 (T_1/2 = 10 h). For three junction positions, the NTCP deviation was statistically significant for T_1/2 longer than 7 h upon changing to the daily plan for a gap size of − 5 mm; the NTCP deviation increased negatively as T_1/2 decreased and reached a maximum of − 0.03 (T_1/2 = 10 h).

We manually calculated the NTCP curves with the IR model by varying EQD₂ which is given to 100% or 1% of the volume of spinal cord (Fig. 5). When EQD₂ was given to the entire spinal cord (Fig. 5(a)), an EQD₂ of 68.6 Gy gave an NTCP of 50% for T_1/2 = 0 (i.e., without the IR model). However, the NTCP curve shifted upward upon incorporating the IR model for T_1/2 longer than ~ 3 h. Moreover, a larger T_1/2 gave a higher NTCP curve. When EQD₂ was given to 1% of the volume of the spinal cord (Fig. 5(b)), the NTCP was not saturated with high EQD₂ because of partial exposure and continued to increase with increasing EQD₂. Although the NTCP was lower than that when the entire spinal cord was irradiated, it increased as T_1/2 increased over ~ 3 h, similar to that when the entire spinal cord was irradiated.

Figure 5 also shows the EQD₂ that gave an NTCP of 5% for various T_1/2, assuming that EQD₂ was given to (c) 100% and (d) 1% of the volume of the spinal cord. In Fig. 5(c), an EQD₂ of 52 Gy gives an NTCP of 5% for T_1/2 = 0, but this value decreases for T_1/2 over 3 h and reaches at least 44 Gy (T_1/2 = 10 h). In Fig. 5(d), an EQD₂ of 58 Gy gives an NTCP of 5% for T_1/2 = 0, but this value decreases for T_1/2 over 3 h and reaches at least 49 Gy (T_1/2 = 10 h).